That's wrong then. The word "soviet" means "council".. To the extent that its etymology connects to "togetherness" it just refers to the group of people coming together to make decisions.
The "union" refers to the union of the socialist republics. You can easily have "soviets" without a union, and you can obviously have a union without soviets.
The "union" refers to the union of the socialist republics. You can easily have "soviets" without a union, and you can obviously have a union without soviets.
Source: born in the USSR